My professor vaguely defined a cluster point and I'm a bit confused on the difference between limit point and cluster point.

He said that

" Suppose a_n converges to alpha.

Sequence {a_n} has a cluster point, if a subsequence of a_n converges to alpha.

If a is a cluster point, it is a limit point."

Can you clarify the difference between cluster point and limit point.

I am currently in a introductory analysis class.

Thank you.

Apr 1st 2013, 03:47 AM

Plato

Re: Limit point and cluster point

Quote:

Originally Posted by rokman54

My professor vaguely defined a cluster point and I'm a bit confused on the difference between limit point and cluster point. He said that
" Suppose a_n converges to alpha.
Sequence {a_n} has a cluster point, if a subsequence of a_n converges to alpha.
If a is a cluster point, it is a limit point."
Can you clarify the difference between cluster point and limit point.
I am currently in a introductory analysis class.

There is a very common misconception here. The limit of a sequence is a cluster point of the sequence but a cluster of a sequence may not be a limit of a sequence. The sequence has two cluster points, but the sequence does not converge so it has no limit although it does have two limit points.

Apr 1st 2013, 05:07 AM

rokman54

Re: Limit point and cluster point

What is the formal definition of cluster point?

Apr 1st 2013, 05:10 AM

Plato

Re: Limit point and cluster point

Quote:

Originally Posted by rokman54

What is the formal definition of cluster point?

Did you look at that web page? Cluster point is just another type of limit point.

Apr 1st 2013, 06:22 AM

HallsofIvy

Re: Limit point and cluster point

The Wikipedia page Plato links to gives the definition of a "cluster point":
A point x ∈ X is a cluster point or accumulation point of a sequence , n ∈ N, if, for every neighbourhood V of x, there are infinitely many natural numbers n such that ∈ V.

It also says "The set of all cluster points of a sequence is sometimes called a limit set." implying that "cluster point" and "limit point" are names for the same thing. However, Plato's point, before, was that "cluster point" or "limit point" of a sequence may not be a limit of that sequence.